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Welcome to our Physics lesson on **Speed and Acceleration of Oscillating Particles in a Wave**, this is the third lesson of our suite of physics lessons covering the topic of **General Equation of Waves**, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

Just like in SHM, we can find the speed of oscillating particles in a wave by taking the first derivative of y-position with respect to the time. Be careful not confuse the oscillating speed of particles (it lies in the y-direction and is not uniform) with the wave speed (which is horizontal and uniform). Thus,

v_{y}(t) = *dy**/**dt*

=*d[y*_{max} × sin(k × x - ω × t + φ) ]*/**dt*

= y_{max} × ω × cos(k × x - ω × t + φ)

=

= y

Likewise, we obtain for the vertical acceleration of oscillating particles,

a_{y} (t) = *dv**/**dt*

=*-d[y*_{max} × ω × cos(k × x - ω × t + φ) ]*/**dt*

= -y_{max} × ω^{2} × sin(k × x - ω × t + φ)

= -ω^{2} × y(t)

=

= -y

= -ω

A wave oscillates according the equation

y(x,t) = 3 × sin(6π × x - 3π × t)

where x and y are in metres and t is in seconds. Calculate:

- Wavelength
- Period of the wave
- Wave's speed
- Vertical position at t = 2 s of a point of wave that oscillates according the vertical line x = 3 m
- Vertical speed of the above point at t = 2 s

**a)** From the general equation of waves

y(x,t) = y_{max} × sin(k × x - ω × t + φ)

And comparing it with the actual equation

y(x,t) = 3 × sin(6π × x - 3π × t)

we extract the following values:

y_{max} = A = 3 m

k = 6π rad/m

ω = 3π rad/s

k = 6π rad/m

ω = 3π rad/s

and

φ = 0

Thus, giving that

k = *2π**/**λ* = 6π

we obtain for the wavelength λ

λ = *2π**/**6π* = *1**/**3* = 0.33 m

**b)** Since

ω = *2π**/**T* = 3π

we obtain for the period

T = *2π**/**3π* = 0.67 s

**c)** Now we can use the simplified equation of wave

v = λ × f = *λ**/**T*

to find the wave speed. Thus,

v = *0.33 m**/**0.67 s* = 0.5 m/s

**d)** Since the vertical position y of a point of wave at any instant t is given by the equation

y(x,t) = y_{max} × sin(k × x - ω × t + φ)

and since in the specific case this equation is

y(x,t) = 3 × sin(6π × x - 3π × t)

we obtain for x = 3 and t = 2

y(3m,2s) = 3 × sin(6π × 3 - 3π × 2)

= 3 × sin12π

= 0

= 3 × sin12π

= 0

This result means the given point of the wave is at the equilibrium position at t = 2 s.

**e)** Vertical speed is the first derivative of vertical position in respect to the time. Thus, we have

v_{y} = *dy**/**dt*

=*d[3 × sin(6π × x - 3π × t) ]**/**dt*

= 3 × 6π × cos(6π × x - 3π × t)

= 18π × cos(6π × x - 3π × t)

=

= 3 × 6π × cos(6π × x - 3π × t)

= 18π × cos(6π × x - 3π × t)

Thus, for x = 3 and t = 2 we obtain for the vertical speed v_{y}

v_{y} (3m,2s) = 18π × cos(6π × 3 - 3π × 2)

= 18π × cos(12π)

= 18 × 3.14 × 1

= 56.52 m/s

= 18π × cos(12π)

= 18 × 3.14 × 1

= 56.52 m/s

You have reached the end of Physics lesson **11.2.3 Speed and Acceleration of Oscillating Particles in a Wave**. There are 3 lessons in this physics tutorial covering **General Equation of Waves**, you can access all the lessons from this tutorial below.

Enjoy the "Speed and Acceleration of Oscillating Particles in a Wave" physics lesson? People who liked the "General Equation of Waves lesson found the following resources useful:

- Speed Feedback. Helps other - Leave a rating for this speed (see below)
- Waves Physics tutorial: General Equation of Waves. Read the General Equation of Waves physics tutorial and build your physics knowledge of Waves
- Waves Revision Notes: General Equation of Waves. Print the notes so you can revise the key points covered in the physics tutorial for General Equation of Waves
- Waves Practice Questions: General Equation of Waves. Test and improve your knowledge of General Equation of Waves with example questins and answers
- Check your calculations for Waves questions with our excellent Waves calculators which contain full equations and calculations clearly displayed line by line. See the Waves Calculators by iCalculator™ below.
- Continuing learning waves - read our next physics tutorial: Energy and Power of Waves

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